
theorem HTh15:
  for V, W being non empty ModuleStr over INT.Ring, f being FrForm of V,W,
  a being Element of F_Real, v being Vector of V holds
  FrFunctionalFAF(a*f,v) = a*FrFunctionalFAF(f,v)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f be FrForm of V,W,
    a be Element of F_Real, w be Vector of V;
    now
      let v be Vector of W;
      thus (FrFunctionalFAF(a*f,w)).v = (a*f).(w,v) by HTh8
      .= a*f.(w,v) by Def3
      .= a*(FrFunctionalFAF(f,w)).v by HTh8
      .= (a* FrFunctionalFAF(f,w)).v by HDef6;
    end;
    hence thesis by FUNCT_2:63;
  end;
